A Rigorous Introduction to Brownian Motion - …

1 A Rigorous Introduction to Brownian MotionAndy DahlAugust 19, 2010 AbstractIn this paper we develop the basic properties of Brownian Motion thengo on to answer a few questions regarding its zero set and its local The Basics12 The Relevant Measure Theory53 Markov Properties of Brownian motion64 Further Properties of Brownian motion91 The BasicsThe concept of a Brownian Motion was discovered when Einstein observedparticles oscillating in liquid. Since fluid dynamics are so chaotic and rapidat the molecular level, this process can be modeled best by assuming theparticles move randomly and independently of their past Motion . We canalso think of Brownian Motion as the limit of a random walk as its timeand space increments shrink to 0. In addition to its physical importance, Brownian Motion is a central concept in stochastic calculus which can beused in finance and economics to model stock prices and interest Brownian Motion DefinedSince we are trying to capture physical intuition, we define a Brownian mo-tion by the properties we want it to have and worry about proving the exis-tence of and explicitly constructing such a process stochastic process{B(t) :t 0}is called ad-dimensionalBrownian motionstarting atx Rdif it has the followingfour properties: Start at x: B(0) = x Independent increments: for all 0 t1 t2.

2 Tn, the incrementsB(tn) B(tn 1),..,B(t2) B(t1)are independent random variables Normality: for allt 0andh >0the incrementB(t+h) B(t)isdistributedN(0,h) Continuity: almost surely,t7 B(t)is continuousThe first property anchors the stochastic process in space. The secondcaptures the continually random nature of a particle that is being constantlybuffeted by fluid molecules. The third is required because the expecteddisplacement of a particle should be proportional to the time it has beentraveling and should be symmetrically distributed about the starting Motion is continuous which explains the fourth a Brownian Motion is frequently denoted{B(t)|t 0}to stress thefact that it is actually an uncountable family of random variables, we willuseBtas shorthand, understanding thattvaries over the non-negative bulk of the first and third sections apply to general Brownian motionsand in the fourth we specialize to the linear construction of Brownian Motion is tedious and beyond the scope ofthis paper.

3 But we should remember that it is the characteristics of Brownianmotion, rather than its construction, which define it. Indeed, there are evendifferent constructions. The details of the construcion will not be used inthis Nondifferentiability of Brownian motionThe most striking quality of Brownian Motion is probably its nowhere surely, Brownian Motion is nowhere differentiableThe proof consists primarily of a long computation which we do notpresent. We will prove later that in any small interval to the right of sometimes,Btattains values greater than and less thanBs. So for all >0 ands 0, we can choose someh (s,s+ ) such thatBs+h Bshis either positiveor negative. This supports the idea that the upper and lower derivatives2ofBtat every point are + and , respectively, although a good deal ofcomputational work goes into proving that the upper and lower limits divergeas 0.

4 Nonetheless, knowing that they do diverge does give us insightinto how rapidly and erratically Brownian Motion jumps theorem can also be understood directly from the definition of Brow-nian Motion . IfBtwere differentiable at some points, we would know whereit was going in some small time interval in the future, but the independentincrement property ofBtshould make us skeptical of such a next proposition is a manifestation of the combination of Brownianmotion s nondifferntiability and its surely,Btis not monotonic on any 0 a < b. LetP(a,b) be the probability thatBtis monotonicon (a,b). Then, by independence of increments,P(a,b) 12 P(a,a+b2) P(a+b2,b) 12 P(a,a+b2)since, even ifBtis monotonic on (a,a+b2) and (a+b2,b), the probability thatit is monotonic in the same direction on both intervals is12. Iterating thedivisions of the interval into halvesntimes we getP(a,b) (12)n P(a,a+b a2n)Taking the limit asn , we seeP(a,b) (12)n P(a,a+b a2n) 12n 0,showingP(a,b) = 0, so any fixed interval is almost surely not taking the countable union over all intervals with rational endpointswe can see thatBtis almost surely not monotonic on any interval withrational endpoints.

5 By the density ofQ R, there is an interval withrational endpoints contained within every interval. So every interval containsa subinterval which is almost surely not monotonic, thus every interval isalmost surely not will use this result later when discussing the maxima Scaling Properties of Brownian MotionWe often study transformations of functions which leave certain propertiesinvariant, and it is natural to ask what transformations ofBthave the 1. Btis a Brownian Motion . Continuity and independence areclearly maintained by negative multiplication and, since the normal distribu-tion is symmetric about zero, all the increments have the proper means now move on to more interesting and useful : IfBtis a standard Brownian Motion , then sois the processXt=aBta2, for alla > and independence of increments still hold. For allt > s 0, the normal random variableX(t) X(s) =a(B(ta2) B(sa2)) is distributedaN(0,t sa2)d=N(0,t s), soX(t) X(s) N(0,t s) as proposition tells us thatBtis a Brownian motions on all time scalesas long as we compensate for the change in variance of the increments bytaking a scalar multiple of the process.

6 More surprisingly, we can invert thedomain ofBtand still have a Brownian : LetBtbe a standard Brownian the processXt={0:t= 0tB1t:t6= 0is also a standard Brownian Brownian motions,Cov(Bt,Bt+s) = Cov(Bt,Bt+s Bt) + Cov(Bt,Bt) =tfor allt,s 0. For our processXt,Cov(Xt,Xt+s) = Cov(tB1t,(t+s)B1t+s)=t(t+s)Cov(B1t,B1t+s ) =t(t+s)1t+s=tSo Cov(Xt,Xt+s Xt) = Cov(Xt,Xt+s) Var(Xt) =t t= 0. Because therandom variablesXt+sandXtare normal, Cov(Xt,Xt+s Xt) = 0 impliesthatXt+s XtandXtare independent. And Var(Xt+s Xt) = Var(Xt+s) +Var(Xt) 2 Cov(Xt+s,Xt) = (t+s) +t 2t=s, so our increments areindependent and have the right is clear fort >0. We know thatXthas the distribution of aBrownian Motion onQ, so0 = limn X(1n) = limt 0X(t)4and we conclude thatXtis continuous att= 0, soXtsatisfies the propertiesof a standard Brownian end with section with an example which demonstrates the computa-tional usefulness of these alternative expressions for Brownian a standard Brownian Motion andXt= astandard Brownian Motion , solimt Xtt= limt B1t=B0= 02 The Relevant Measure TheoryWe assume the reader is familiar with the elements of basic probability theorysuch as expectation, covariance, normal random variables, etc.}

7 But we do addrigor to these notions by developing the underlying measure theory, whichwill be necessary for our discussion of the Markov -algebra on a setSis a subset of2S, where2 Sis thepower set ofS, satisfying: { } for allA ,Ac for all sequencesA0,A1,.. , i=0Ai By de Morgan s laws we can see that -algebras are closed under count-able intersections as well. The -algebra will be our object of measurement,so now we need to develop our method of a countably additive map : 7 [0, ], where 2 Sis our -algebra on some setS. A countably additive map is onesuch that for any sequenceA1,A2,.. of disjoint events, ( i=1Ai) = i=1 (Ai). A probability measure is a measure such that (S) = definition implies that ({ }) = 0 because ({ }) = ({ } { }) = ({ }) + ({ }). Our next definition collects these tripleis the triple( ,F,P)whereFis a -algebra on the set andP: 7 [0,1]is a probability measure.

8 We call the sample space andFthe collection of (P-measurable) triple provides the background for the study probability. In theforeground are random variableis anF-measurable mapX: 7 R,meaning that the preimageX 1(B) Ffor allB B(R). The law ofXisP(X 1) :B(R)7 [0,1].The random variableXis a correspondence between events and sets inR, which formalizes the notion thatXtakes on certain values when certainevents occur. Of course, this correspondence is not that interesting in itself;what interests us is the probability ofXlying in sets inR, which is given bythe law ofX. For the law ofXto be well defined we needX 1(B(R) F,sinceFis the domain ofP, which is why we requireXto stochastic process is a family of random variables that evolves overtime, and up to this point we have viewed these random variables from time0. But we can also look at the process at some timesat which the set{Xt|0 t s}is known, and the probability of events occuring pastswilldepend on this a probability space( ,F,P)is a family{Ft|t 0}of -algebras such thatFs Ft Ffor alls < t.)

9 A stochasticprocess{Xt|t 0}is adapted to the filtration ifXtisFtmeasurable for allt adapted filtration captures the intuition of our information whichevolves along with our process: our information grows as time goes Markov Properties of Brownian motionThe Markov properties tell us at what timessa Brownian Motion {Bt+s|t 0} referred to asBt+sin the future has the same distribution as a Brown-ian Motion started atBsor, alternatively, when the processBt+s Bsis astandard Brownian Motion . We will refer to this phenomenon as Brownianmotion starting anew at times. Our independence of increments require-ment might seem to make this property trivial, and, for deterministic times,it 2.(Markov Property)LetBtbe a Brownian Motion and fixs +s Bsis a standard Brownian Motion independent of{Bt|0 t s}. is clear thatBt+sis a Brownian Motion . Subtracting a constantonly changes the starting point, and, in particular, subtractingBsmakesthe process a standard Brownian Motion .

10 Independence ofBtbefore timesfollows from the independence of increments of Brownian far more interesting and important class of times is random times,meaning times defined by some randomly occurring event. Brownian motiondoes not necessarily start afresh at such times. We provide an example, butfirst state a continuous functionfis said to attain amaximumon anintervalIats Iiff(s) f(t)for allt IWe sayfattains alocal maximumatsif there exists a non-degenerateintervalIcontainingson whichf(s)is a maximum. We say the (local)maximum is strict if the above inequality can be replaced by a strict a time thatBtattains a strict local maximum and defineXt=Bt+s Bs. Then there exists some such that for allr (s ,s+ ),Br< Bs. SoP(X 2 X0<0) =P(Bs+ 2 Bs>0) = 0. So the incrementX 2 X0<0is certainly not normal, thusXtis not a Brownian Motion andBtdoes not start anew example shows we need to be careful when considering random , Brownian Motion does start anew at some random random timeT [0, ]defined on a probability space withfiltrationFtis astopping timeif{T s} Fsfor everys > our heuristic understanding ofFtas the information up to timet, arandom time is a stopping time if we can determine whether it has occurredbeforesbased only on knowing the information up tos.